Subgroups — C₃₁×#8920 #21465679

Order 7936.

644 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2366	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#1957	C₂	1	1	yes	maximal, characteristic
3	C₃₁×#1947	C₂	1	2	yes	maximal, complemented
4	C₂×C₃₁×#331	C₂	1	1	yes	maximal, characteristic
5	C₃₁×#1279	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
7	C₂×C₄×D₄×C₃₁	—	2	2	no	—
8	C₂×C₃₁×#107	—	2	2	no	—
9	C₂×C₃₁×C₄⋊C₈	C₂²	1	2	yes	—
10	C₂×C₃₁×#102	—	2	2	no	—
11	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₃₁×#390	—	2	2	no	—
13	C₃₁×#386	—	2	2	no	—
14	C₃₁×#384	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
16	C₃₁×#340	—	2	2	no	—
17	C₃₁×#331	C₄	1	4	yes	complemented
18	C₂²×D₄×C₃₁	—	2	2	no	—
19	C₂²×D₄×C₃₁	D₄	1	2	yes	—
20	C₂³×C₄×C₃₁	—	2	2	no	—
21	C₂²×C₈×C₃₁	—	2	1	no	—
22	C₂²×C₈×C₃₁	—	2	2	no	—
23	C₃₁×#127	—	2	2	no	—
24	C₃₁×#127	C₂×C₄	1	2	yes	—
25	C₃₁×#127	Q₈	1	2	yes	—
26	C₄×D₄×C₃₁	—	4	4	no	—
27	C₄×D₄×C₃₁	—	4	4	no	—
28	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
29	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
30	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
32	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
33	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
34	C₂×C₄²×C₃₁	—	2	2	no	—
35	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
36	C₃₁×#107	—	4	4	no	—
37	C₃₁×C₄⋊C₈	—	2	2	no	—
38	C₃₁×C₄⋊C₈	C₂×C₄	1	4	yes	complemented
39	C₃₁×#102	—	4	4	no	—
40	C₄×C₈×C₃₁	—	2	2	no	—
41	C₃₁×#95	—	2	2	no	—
42	C₂⁴×C₃₁	—	2	2	no	—
43	C₂×D₄×C₃₁	—	2	4	no	—
44	C₂×D₄×C₃₁	—	2	4	no	—
45	C₂×D₄×C₃₁	—	2	2	no	—
46	C₂×D₄×C₃₁	—	4	2	no	—
47	C₂×D₄×C₃₁	—	4	2	no	—
48	C₂×D₄×C₃₁	—	4	2	no	—
49	C₂×D₄×C₃₁	—	4	2	no	—
50	C₂×D₄×C₃₁	—	4	2	no	—
51	C₂×D₄×C₃₁	—	4	2	no	—
52	C₂×D₄×C₃₁	C₄⋊C₄	1	4	yes	—
53	C₂²×C₄×C₃₁	—	2	2	no	—
54	C₂²×C₄×C₃₁	—	2	2	no	—
55	C₂²×C₄×C₃₁	—	4	8	no	—
56	C₂²×C₄×C₃₁	—	4	1	no	—
57	C₂²×C₄×C₃₁	—	8	2	no	—
58	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₃₁	—	2	2	no	—
61	C₂×C₈×C₃₁	—	2	4	no	—
62	C₂×C₈×C₃₁	—	4	1	no	—
63	C₂×C₈×C₃₁	—	4	1	no	—
64	C₂×C₈×C₃₁	—	4	2	no	—
65	C₂×C₈×C₃₁	—	4	2	no	—
66	C₄⋊C₄×C₃₁	—	2	2	no	—
67	C₄⋊C₄×C₃₁	—	2	2	no	—
68	C₄⋊C₄×C₃₁	—	2	2	no	—
69	C₄⋊C₄×C₃₁	—	4	2	no	—
70	C₄⋊C₄×C₃₁	—	4	2	no	—
71	C₄⋊C₄×C₃₁	—	4	2	no	—
72	C₄⋊C₄×C₃₁	D₈	1	4	yes	—
73	C₂²⋊C₄×C₃₁	—	4	8	no	—
74	C₂²⋊C₄×C₃₁	—	8	2	no	—
75	C₄²×C₃₁	—	4	2	no	—
76	C₄²×C₃₁	—	4	2	no	—
77	C₄²×C₃₁	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₃₁	C₂×Q₈	1	1	yes	characteristic
79	C₄²×C₃₁	#41	1	2	yes	—
80	#8920	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
81	C₂³×C₃₁	—	2	4	no	—
82	C₂³×C₃₁	—	4	4	no	—
83	C₂³×C₃₁	—	4	4	no	—
84	C₂³×C₃₁	—	4	2	no	—
85	C₂³×C₃₁	—	4	2	no	—
86	C₂³×C₃₁	#120	1	1	yes	characteristic
87	D₄×C₃₁	—	4	4	no	—
88	D₄×C₃₁	—	4	4	no	—
89	D₄×C₃₁	—	4	4	no	—
90	D₄×C₃₁	—	4	4	no	—
91	D₄×C₃₁	—	4	4	no	—
92	D₄×C₃₁	—	8	2	no	—
93	C₂×C₄×C₃₁	—	2	1	no	—
94	C₂×C₄×C₃₁	—	2	1	no	—
95	C₂×C₄×C₃₁	—	2	4	no	—
96	C₂×C₄×C₃₁	—	4	2	no	—
97	C₂×C₄×C₃₁	—	4	2	no	—
98	C₂×C₄×C₃₁	—	4	2	no	—
99	C₂×C₄×C₃₁	—	4	2	no	—
100	C₂×C₄×C₃₁	—	4	2	no	—
101	C₂×C₄×C₃₁	—	8	8	no	—
102	C₂×C₄×C₃₁	—	8	2	no	—
103	C₂×C₄×C₃₁	—	8	1	no	—
104	C₂×C₄×C₃₁	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₃₁	C₂²⋊Q₈	1	2	yes	—
107	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
108	C₂×C₄×C₃₁	C₂×D₈	1	2	yes	—
109	C₂×C₄×C₃₁	#135	1	2	yes	—
110	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
111	C₂×C₄×C₃₁	#137	1	2	yes	—
112	C₈×C₃₁	—	4	2	no	—
113	C₈×C₃₁	—	4	4	no	—
114	#2366	C₂×C₃₁	1	1	yes	characteristic
115	#1957	C₂×C₃₁	1	1	yes	characteristic
116	#1947	C₂×C₃₁	1	2	yes	complemented
117	C₂×#331	C₂×C₃₁	1	1	yes	characteristic
118	#1279	C₂×C₃₁	1	2	yes	complemented
119	C₂²×C₃₁	—	4	4	no	—
120	C₂²×C₃₁	—	4	8	no	—
121	C₂²×C₃₁	—	8	2	no	—
122	C₂²×C₃₁	—	8	2	no	—
123	C₂²×C₃₁	—	8	4	no	—
124	C₂²×C₃₁	#385	1	1	yes	characteristic
125	C₂²×C₃₁	#447	1	2	yes	—
126	C₂²×C₃₁	#452	1	2	yes	—
127	C₂²×C₃₁	#453	1	1	yes	characteristic
128	C₂²×C₃₁	#455	1	1	yes	characteristic
129	C₄×C₃₁	—	2	1	no	—
130	C₄×C₃₁	—	2	1	no	—
131	C₄×C₃₁	—	4	4	no	—
132	C₄×C₃₁	—	8	2	no	—
133	C₄×C₃₁	—	16	1	no	—
134	C₄×C₃₁	C₄×D₈	1	2	yes	—
135	C₄×C₃₁	#440	1	2	yes	—
136	C₄×C₃₁	#474	1	2	yes	—
137	C₄×C₃₁	#478	1	2	yes	—
138	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
139	C₂×C₄×D₄	—	2	2	no	—
140	C₂×#107	—	2	2	no	—
141	C₂×C₄⋊C₈	C₂²×C₃₁	1	2	yes	—
142	C₂×#102	—	2	2	no	—
143	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
144	#390	—	2	2	no	—
145	#386	—	2	2	no	—
146	#384	C₂²×C₃₁	1	1	yes	characteristic
147	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
148	#340	—	2	2	no	—
149	#331	C₄×C₃₁	1	4	yes	complemented
150	C₂×C₃₁	—	8	4	no	—
151	C₂×C₃₁	#1619	1	1	yes	characteristic
152	C₂×C₃₁	#1630	1	1	yes	characteristic
153	C₂×C₃₁	#1641	1	2	yes	—
154	C₂×C₃₁	#1856	1	1	yes	characteristic
155	C₂×C₃₁	#1862	1	2	yes	—
156	C₂²×D₄	—	2	2	no	—
157	C₂²×D₄	D₄×C₃₁	1	2	yes	—
158	C₂³×C₄	—	2	2	no	—
159	C₂²×C₈	—	2	2	no	—
160	C₂²×C₈	—	2	1	no	—
161	#127	—	2	2	no	—
162	#127	C₂×C₄×C₃₁	1	2	yes	—
163	#127	Q₈×C₃₁	1	2	yes	—
164	C₄×D₄	—	4	4	no	—
165	C₄×D₄	—	4	4	no	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	—	2	1	no	—
168	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
169	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
170	C₂×C₂²⋊C₄	—	2	4	no	—
171	C₂×C₂²⋊C₄	—	2	2	no	—
172	C₂×C₄²	—	2	2	no	—
173	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
174	#107	—	4	4	no	—
175	C₄⋊C₈	—	2	2	no	—
176	C₄⋊C₈	C₂×C₄×C₃₁	1	4	yes	complemented
177	#102	—	4	4	no	—
178	C₄×C₈	—	2	2	no	—
179	#95	—	2	2	no	—
180	C₃₁	#8920	1	1	yes	characteristic, direct product factor, Sylow
181	C₂⁴	—	2	2	no	—
182	C₂×D₄	—	2	4	no	—
183	C₂×D₄	—	2	4	no	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	—	4	2	no	—
186	C₂×D₄	—	4	2	no	—
187	C₂×D₄	—	4	2	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂×D₄	—	4	2	no	—
190	C₂×D₄	—	4	2	no	—
191	C₂×D₄	C₄⋊C₄×C₃₁	1	4	yes	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	4	8	no	—
195	C₂²×C₄	—	4	1	no	—
196	C₂²×C₄	—	8	2	no	—
197	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
198	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	—	4	1	no	—
203	C₂×C₈	—	4	2	no	—
204	C₂×C₈	—	4	2	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	2	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	—	4	2	no	—
211	C₄⋊C₄	D₈×C₃₁	1	4	yes	—
212	C₂²⋊C₄	—	4	8	no	—
213	C₂²⋊C₄	—	8	2	no	—
214	C₄²	—	4	2	no	—
215	C₄²	—	4	2	no	—
216	C₄²	C₂²×C₄×C₃₁	1	1	yes	characteristic
217	C₄²	C₂×Q₈×C₃₁	1	1	yes	characteristic
218	C₄²	#41×C₃₁	1	2	yes	—
219	C₂³	—	2	4	no	—
220	C₂³	—	4	4	no	—
221	C₂³	—	4	4	no	—
222	C₂³	—	4	2	no	—
223	C₂³	—	4	2	no	—
224	C₂³	C₃₁×#120	1	1	yes	characteristic
225	D₄	—	4	4	no	—
226	D₄	—	4	4	no	—
227	D₄	—	4	4	no	—
228	D₄	—	4	4	no	—
229	D₄	—	4	4	no	—
230	D₄	—	8	2	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	8	8	no	—
240	C₂×C₄	—	8	2	no	—
241	C₂×C₄	—	8	1	no	—
242	C₂×C₄	C₂×C₄⋊C₄×C₃₁	1	1	yes	characteristic
243	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
244	C₂×C₄	C₃₁×C₂²⋊Q₈	1	2	yes	—
245	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
246	C₂×C₄	C₂×D₈×C₃₁	1	2	yes	—
247	C₂×C₄	C₃₁×#135	1	2	yes	—
248	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
249	C₂×C₄	C₃₁×#137	1	2	yes	—
250	C₈	—	4	2	no	—
251	C₈	—	4	4	no	—
252	C₂²	—	4	4	no	—
253	C₂²	—	4	8	no	—
254	C₂²	—	8	2	no	—
255	C₂²	—	8	2	no	—
256	C₂²	—	8	4	no	—
257	C₂²	C₃₁×#385	1	1	yes	characteristic
258	C₂²	C₃₁×#447	1	2	yes	—
259	C₂²	C₃₁×#452	1	2	yes	—
260	C₂²	C₃₁×#453	1	1	yes	characteristic
261	C₂²	C₃₁×#455	1	1	yes	characteristic
262	C₄	—	2	1	no	—
263	C₄	—	2	1	no	—
264	C₄	—	4	4	no	—
265	C₄	—	8	2	no	—
266	C₄	—	16	1	no	—
267	C₄	C₄×D₈×C₃₁	1	2	yes	—
268	C₄	C₃₁×#440	1	2	yes	—
269	C₄	C₃₁×#474	1	2	yes	—
270	C₄	C₃₁×#478	1	2	yes	—
271	C₂	—	8	4	no	—
272	C₂	C₃₁×#1619	1	1	yes	characteristic
273	C₂	C₃₁×#1630	1	1	yes	characteristic
274	C₂	C₃₁×#1641	1	2	yes	—
275	C₂	C₃₁×#1856	1	1	yes	characteristic
276	C₂	C₃₁×#1862	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	#120
commutator	C₄²	C₂²×C₄×C₃₁
Frattini, cosocle	C₂×C₄²	C₂³×C₃₁
prime radical	C₂×C₃₁×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₃₁×#1947	C₂	semidirect
5. C₃₁×#1279	C₂	semidirect
17. C₃₁×#331	C₄	semidirect
38. C₃₁×C₄⋊C₈	C₂×C₄	semidirect
80. #8920	C₃₁	direct (N × H)
116. #1947	C₂×C₃₁	semidirect
118. #1279	C₂×C₃₁	semidirect
149. #331	C₄×C₃₁	semidirect
176. C₄⋊C₈	C₂×C₄×C₃₁	semidirect
180. C₃₁	#8920	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#2366	2
C₃₁×#1957	2
C₃₁×#1947	2
C₂×C₃₁×#331	2
C₃₁×#1279	2
#8920	31

Subgroups — C31×#8920 #21465679

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#8920 #21465679